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Introduction to Functional Analysis

Introduction to Functional Analysis

Code	13933
Year	3
Semester	S1
ECTS Credits	6
Workload	TP(60H)
Scientific area	Mathematics
Entry requirements	NA
Learning outcomes	In this curricular unit are introduced the basic concepts of measurement and integration theory and the Banach and Hilbert spaces. At the end of the curricular unit the students should be able to: - define and use measure space; - find out if a function is measurable; - compute the Lebesgue integral and apply its main properties; - use the L^p spaces; - identify the main normed spaces; - apply the theory of the normed spaces and of the Banach spaces; - identify the main spaces with internal product; - apply the theory of spaces with an inner product and of Hilbert spaces; - know and use the main theorems of functional analysis: Hahn-Banach, Banach-Steinhaus, open application and closed graph.
Syllabus	1. Measure and Integration 1.1 Semi-algebras, algebras e s-algebras 1.2 Measures 1.3 Caratheodory’s extension theorem 1.4 Measurable functions 1.5 Lebesgue’s integral 1.6 Monotone convergence and dominated convergence theorems 1.7 L^p spaces 1.8 Product measure and Fubini’s theorem 2. Normed spaces and Banach spaces 2.1 Definition, elementary properties and examples 2.2 Continuous linear operators 2.3 Functionals and dual space 2.4 Finite-dimensional Banach spaces 2.5 Compactness and Riesz Lemma 3. Inner product spaces and Hilbert spaces 3.1 Definition, elementary properties and examples 3.2 Orthogonal complement and ortogonal projections 3.3 Orthonormal sets 3.4 Functionals in Hilbert spaces 3.5 Adjoint operator 4. Fundamental theorems of Functional Analysis 4.1 Zorn’s Lemma 4.2 Hahn-Banach theorem 4.3 Dual operator 4.4 Reflexive spaces 4.5 Banach-Steinhaus theorem 4.6 Open mapping theorem and closed graph theorem
Main Bibliography	Bollobás, B. (1999). Linear Analysis: An Introductory Course. Cambridge University Press. 2th edition Conway, J. B. (2013). A course in functional analysis. Springer Science & Business Media. de Castro Jr, A. A. (2015). Curso de teoria da medida. Instituto de Matemática Pura e Aplicada. 3.ª edição Fernandez, P. J. (2015). Medida e integração. Instituto de Matemática Pura e Aplicada. 2.ª edição Giles, J. R., & Giles, J. R.(2000). Introduction to the analysis of normed linear spaces. Cambridge University Press. Kreyszig, E. (1978). Introductory functional analysis with applications New York: Wiley. Michel, A. N., & Herget, C. J. (2009). Algebra and analysis for engineers and scientists. Springer Science & Business Media. Rynne, B., & Youngson, M. A. (2011). Análise Funcional Linear. Coleção Ensino da Ciência e Tecnologia. IST Press. Taylor, A. E., & Lay, D. C. (1986). Introduction to functional analysis. Krieger Publishing Oliveira, César R. (2015). Introdução à Análise Funcional. IMPA
Language	Portuguese. Tutorial support is available in English.